Real Analysis IIPlease find related sample as guide to solve above problem 1. Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) where \( f(x) = x^2 \), \( c = -1 \), and \( u = 2 \). Use the \( \epsilon \)-\( \delta \) definition to prove \( L_u = ? \) (you need to find \( L_u \) in your scratchwork). **Example:**Let \( f: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined as \( f(x, y) = \left( x^2 y, (3x + 2)y \right) \). Consider vectors \( u = (1, 0) \) and \( c = (1, 1) \).**Claim:** \( Lu = (2, 3) \)**Proof:**Fix \(\epsilon > 0\). Choose \(\delta = \frac{\epsilon}{2}\). Assume \(0 < |t| < \delta\). Then:\[\frac{1}{t} \left[ f(c + tu) - f(c) - Lu \right] = \left \lVert \frac{1}{t} \left[ f(1+t, 1) - (1, 5) - (2, 3) \right] \right \rVert\]\[= \left \lVert \frac{1}{t} \left[ ((1+t)^2, 3(1+t)+2) - (1, 5) \right] - (2, 3) \right \rVert\]\[= \left \lVert \frac{1}{t} \left[ (1 + 2t + t^2, 3t + 5) - (1, 5) \right] - (2, 3) \right \rVert \]\[= \left \lVert \frac{1}{t} \left[ (2t + t^2, 3t) - (2, 3) \right] \right \rVert \]\[= \left \lVert \frac{1}{t} \left[ (t^2, 0) \right] \right \rVert \]\[= \left \lVert (t, 0) \right \rVert = |t| < \delta = \epsilon\]**Why is \( Lu = (2, 3) \)?**\[L = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\

Answered: 1. Let ƒ : R → R where ƒ(x) = x², c =…

1. Let ƒ : R → R where ƒ(x) = x², c = -1, and u = 2. Use the e-d definition to prove Lu=? (you need to find Lu in your scratchwork).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Part 1 - Estimate the mean price of regular gasoline in the Dayton area by taking a sample. What is the population you will be studying? Think carefully about how you are going to get a good sample to calculate a sample mean and collect the sample. For example, driving around your neighborhood is an example of a convenience sample and should not be done. Think of a more efficient way to collect gas prices that will also cover the whole Dayton area. Write a paragraph or two explaining how you collected your sample and why you think your sample is representative of Dayton gas prices. What is the sample mean? Construct a 95% confidence interval for the population mean price of gasoline in the Dayton area. Explain what this confidence interval tells you. Part 2 - Compare your sample estimate to the current mean gas price in the State of Ohio. Find the mean price of gasoline in the State of Ohio. You might try this GasBuddy website. Set up a null and alternative hypothesis to see if…
please show all steps and please write out complete SENTENCES as to how you did each step. And explaining what you did to get the answer!!! Thanks
Explain about Other Multiple Comparison Tests.
Below is a research question, please answer the following question. After it is solved, you can simply replace the highlighted sentences (the format you should follow, please follow the highlighted sentence format) with the correct answer.
In your own words, explain three differences between correlation and regression analysis
Discuss how sampling error and standard error might arise within the context of an original real world application